1. Introduction

In modern optical lithography increasingly small structures have to be realized. This is done by reducing the wavelength, increasing the numerical aperture, and by some other means such as a special type of illumination or phase shift masks. The numerical aperture itself NA=n˙ sinφ can be increased by increasing either the half aperture angle φ or the refractive index n of the medium in front of the target. The latter method is used in modern immersion DUV-lithography at a vacuum wavelength of λ=193 nm by putting water between the last surface of the lithography objective and the wafer. The refractive index of water is about n=1.44 at this wavelength reducing the effective value λ/n to about 134 nm. That is smaller than the 157 nm in air, formerly thought to be the next lower wavelength on the roadmap of lithography. Increasing the half aperture angle φ has the natural limit of φ=π/2, i.e. sinφ=1.0.

However, it is well known that the point spread function (PSF) of a linearly polarized plane wave focused by a high numerical aperture objective to a tight spot is no longer rotationally symmetric [1–6] because polarization effects break the symmetry. Consequently, the modulation transfer function (MTF) for incoherent illumination which is the modulus of the inverse Fourier transform of the PSF, will also not be rotationally symmetric. On the other hand, the PSFs for circularly and radial polarization [7] are both rotationally symmetric, and hence the MTFs will be rotationally symmetric in these cases as well. Thus it is very important for modern lithography that we investigate the influence polarization effects have on the PSF and the MTF [8–12]. In this paper we will use the MTF for incoherent illumination, which is of course an approximation because the illumination in lithography is partially coherent. This approximation can still give us at least an idea of which polarization state is the most appropriate at which aperture angle.

In the next section we describe the calculation of the modulation transfer function for different polarization states in theory. In section 3 we deal with the numerical calculation of the MTFs for different polarization states, different aperture angles, and different apodization. In section 4 we discuss the results. In section 5 we give some ideas of how to apply these results to lithography and microscopy. Section 6 is the conclusion of the paper.

2. Calculation of the modulation transfer function

In the following, we assume that the optical imaging system is aplanatic and hence fulfilling the sine condition [13]. Moreover, we assume an ideal lens without aberrations which is ideally anti-reflection coated so that the transmittance is one (or at least constant) at all points of the aperture. The PSF for object points which are not far away from the optical axis will be just a laterally shifted copy of the on-axis PSF because of the aplanatism. The calculation of the PSF is explained in the appendix. In the case of incoherent illumination the intensity distribution in the image plane is then a convolution of the intensity distribution in the object plane and of the PSF. From the imaging of extended objects [13,14] it is well-known that in this case the modulation transfer function (MTF) for incoherent illumination is the modulus of the optical transfer function (OTF), which is the inverse Fourier transform of the point spread function (PSF) apart from a normalization constant:

Here, x,y are the coordinates in the image plane and ν_x,ν_y are spatial frequencies in x- and y-direction. The denominator normalizes the MTF and ensures that the MTF has the value 1 at the spatial frequency ν _x=ν _y=0. This has to be the case because the meaning of the MTF is that it gives the deterioration of the contrast if a grating-like object with a sinusoidal intensity variation and spatial frequencies ν _x,ν _y is imaged. A grating with spatial frequencies zero, i.e. with infinite period, is of course always imaged perfectly so that the MTF has to be 1 at zero spatial frequency.

In the scalar approximation it is also quite easy to show that the MTF is proportional to the modulus of the autocorrelation function of the pupil function P(x′,y′) = A(x′,y′)exp(2πiW(x′,y′)/λ), with the amplitude distribution A and the wave aberrations W expressed as optical path length differences in the exit pupil with coordinates x′ and y′ [14]. It is also well-known from mathematics how to calculate the autocorrelation function of the pupil function at a certain spatial frequency. Two copies of the pupil function are laterally shifted relative to each other and then one calculates the integral of the product of the pupil function with the complex conjugated shifted copy of the pupil function. Since the lateral shift depends on the spatial frequency and the pupil function is zero outside of the aperture there is a maximum spatial frequency where the MTF has a function value different from zero. This maximum frequency is the so called cut-off frequency ν_cut with:

For higher spatial frequencies the MTF is always zero because there is no overlap of the two laterally shifted copies of the pupil function in the calculation of the autocorrelation function.

To calculate the MTF by taking into account polarization effects the PSF has of course to be calculated according to the method described in Ref. [1] as an interference pattern of plane waves traveling along the direction of the geometric rays from the exit pupil to the focus. Therefore, the PSF is proportional to the sum of the electric energy densities of all three components of the electric field in the focus. However, also in this case a kind of pupil function can be defined for each component of the electric field and the MTF will be proportional to the modulus of the sum of the autocorrelation functions of these three pupil functions. Therefore, it is clear that also in the vectorial case of taking into account polarization the maximum spatial frequency for which the MTF can be different from zero is the cut-off frequency defined by Eq. (2).

3. MTFs for different polarization states and numerical apertures

The MTF is calculated numerically with the method of the last section for different polarization states, different aperture angles, and different amplitude functions in the entrance pupil. To avoid aliasing effects by using a Fast Fourier transformation and to have a well-sampled MTF afterwards, the diameter of the PSF data has to be high enough. We used a PSF field with a diameter of D=32λ/NA and 512x512 samples. So, the lateral sampling distance between two points was just ∆x=λ/(16NA) corresponding to a maximal spatial frequency ν _max=1/(2∆x)=8NA/λ. This is 4 times larger than the cut-off frequency of the MTF of Eq. (2) so that we did not explicitly use the fact that the MTF is zero outside of the cut-off frequency, but we can prove it in this way also numerically. Of course, by knowing that the MTF is zero outside of the cut-off frequency, there would also be no aliasing effects if we would take a larger lateral sampling distance of ∆x=λ/(4NA), which is the largest allowed sampling distance (or smallest sampling density) without getting aliasing effects. At the end, we have using our small sampling distance still 128 samples of the MTF with values different from zero, so that the lateral resolution of the MTF is high enough. Additionally, in all our calculations a refractive index of one was assumed so that we have NA=sinφ. This means that the largest influence of the polarization effects onto the PSF and therefore also onto the MTF will be for the limiting case of NA=1.0. Nevertheless, it has to be mentioned that the polarization effects depend on the aperture angle and therefore on sinφ, but not directly on the NA itself which also depends on the refractive index n of the medium. Only, for a non-immersion optical system with a refractive index of one we can use the parameters NA and sinφ with the same meaning.

Normally detectors are sensitive to all components of the electric field so that the MTF has to be calculated by using the complete PSF being proportional to the sum of the electric energy densities of all electric field components. But, as mentioned in Ref. [4], there may also be a special detector which is only sensitive to a certain component of the electric field. By coating a very thin photo resist layer (much smaller than the wavelength of the used light) on a metal layer with high conductivity one can imagine that the lateral field components will vanish in the near field shortly above the metal layer since in the metal layer electric currents are induced which generate reverse lateral electric field components. So, only the longitudinal electric field component will exist in the thin photo resist layer and will be detected. Another possibility would be to use anisotropic molecules which are only sensitive to one field component [15]. If these molecules can additionally be aligned with their symmetry axis in one direction they would form a detector which is only sensitive to one special electric field component. So, it would perhaps also be possible to build a detector for one of the lateral field components only. Of course, especially this second case is in the moment of speculative nature since it is not easy to find such molecules with a high sensitivity to only one component of the electric field and which additionally can be aligned with their axes in one common direction, but parallel to the substrate. However, let us assume below that we have besides a normal detector which is sensitive to all components of the electric field also one detector which is only sensitive to the longitudinal component (in the following named z-component since the optical axis is along the z-axis) and also one detector which is sensitive to one of the lateral components (in the following always the y-component of the electric field).

In the following, it is assumed that a plane wave with a certain polarization state (linear, circular or radial) is focused by a high NA microscope objective which fulfills the sine condition. In the case of linear or circular polarization we assume a constant amplitude in the entrance pupil, whereas for radial polarization first a doughnut shaped electric field distribution

with a beam waist parameter w ₀=0.95 r _aperture (r _aperture= radius of entrance pupil) is taken.

For linear polarization it is well-known [4, 6] that the electric field component in the focus parallel to the direction of linear polarization of the incident light forms a nearly circular spot similar to that assumed in the scalar case, whereas the longitudinal component with its non-rotationally symmetric shape broadens the spot to an elliptic shape. Of course, for very high aperture angles with sinφ near 1.0 also the spot for the electric field component parallel to the direction of polarization of the incident light will be elliptic to some degree. So, we will calculate for linear polarization, which we define to be in the y-direction, the MTFs for the two cases of using all electric field components or for using only the y-component of the electric field.

For circular polarization we will only calculate the MTF for using all components of the electric field.

For radial polarization [4, 7] the z-component (longitudinal component) of the electric field forms in the focus a tight spot with a small maximum and reasonably low side lobes, whereas the lateral components broaden the spot (especially for small values sinφ). Therefore, for radial polarization we will calculate the MTFs for the two cases of using all components of the electric field or for using only the z-component.

Since for circular and radial polarization the PSF and therefore also the MTF is rotationally symmetric, only one section of the MTF will be displayed in this case. For linear polarization the PSF is no longer rotationally symmetric for high values sinφ and therefore for linear polarization we will always display a section of the MTF with spatial frequency ν _x in x-direction, i.e. the grating lines are in this case in y-direction parallel to the direction of polarization of the incident light, and a section with spatial frequency ν _y in y-direction, i.e. the grating lines are here in x-direction perpendicular to the direction of linear polarization.

Figure 1 shows the MTF curves for a full circular aperture and values of sinφ of 0.2, 0.7, 0.8, 0.9, and 1.0. The same is done for an annular aperture with an inner radius of 90% of the aperture radius and Fig. 2 shows the corresponding MTF curves. In both figures, there are for the case of linear polarization curves for ν=ν _x (signated with “x-section”) and for ν=ν _y (signated with “y-section”). Additionally, for linear and radial polarization there are curves where all components of the electric field were taken into account for the PSF (signated with “all components”) or only one component (signated with “y-component only” for linear polarization or “z-component only” for radial polarization).

 

Fig. 1. MTF curves for a full circular aperture and sinφ ranging from 0.2 to 1.0 for different polarization states (linear, circular and radial).

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Fig. 2. MTF curves for an annular aperture (inner radius is 90% of the outer radius) and sinφ ranging from 0.2 to 1.0 for different polarization states (linear, circular and radial).

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Fig. 3. MTF curves for the case (i) of section 3, i.e. radial polarization and different annular apertures.

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Another interesting case is to show the influence of apodization effects in the entrance pupil onto the MTF. We calculated it for the case of radial polarization at the limiting case of sinφ=1.0. Three cases were considered.

Here, two different cases for the waist parameter w ₀ were simulated. (iia) The waist parameter of the Gaussian function is assumed to be constant with w ₀=0.95 r _aperture. (iib) The waist parameter is chosen in such a way that there is the maximum of the electric field amplitude of Eq. (5) exactly at the rim of the aperture:

Remark: There is for n≠0 really a maximum of |E̱| at the position of w ₀ (and not a minimum or saddle point) since |E̱| is a non-negative function with |E̱|=0 at r=0. So, |E̱| increases for r>0 and the only extreme value which exists has to be a maximum. For n=0 Eq. (6) is not valid, but there is a maximum at r=0 since the function is then just a Gaussian function exp(-r ²/w ₀ ²).

 

Fig. 4. MTF curves for the case (iia) of section 3, i.e. radial polarization and different apodization of the aperture with the parameter n ranging from n=0 to n=9. The respective amplitude function |E(r)| is displayed in the figure captions.

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Fig. 5. MTF curves for the case (iib) of section 3, i.e. radial polarization and different apodization of the aperture with the parameter n ranging from n=1 to n=9. The respective amplitude function |E(r)| is displayed in the figure captions.

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Fig. 6. Comparison of some MTF curves using either a “hard mask” apodization via an annular aperture (green curve for r _in/r _aperture=0.6) or a smooth apodization via different amplitude functions (red and black curves with the amplitude function |E| in the figure caption).

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The MTF curves of the case (i) with r _in/r _aperture ranging from 0 (full aperture) to 0.9 are shown in Fig. 3. For the case (iia) and the power n ranging from n=0 to n=9 Fig. 4 gives the MTF curves. Finally, Fig. 5 shows case (iib) with n ranging from n=1 to n=9. In each of the figures the MTF is calculated using either the total electric energy density in the focus or only the z-component.

4. Evaluation of the calculated modulation transfer functions

4.1 Full circular aperture

First, the MTF curves for the full circular aperture of Fig. 1 will be discussed for the different values sinφ and different polarization states. It can easily be seen that for the small value sinφ=0.2 which approaches the scalar case the MTF curves for linear polarization and circular polarization coincide in x- and y-direction independent whether all electric field components are taken or only the y-component. On the other side, the MTF curves for radial polarization are totally different. If the sum of the electric energy density of all components is taken the green short dashed curve results which has a zero of the contrast for a spatial frequency of about 0.9 NA/λ with NA=0.2. So, for higher spatial frequencies there is a contrast inversion which is of course quite bad for optical imaging. On the other side, if only the electric energy density coming from the z-component is taken (long dashed black line), there is a quite high contrast for high spatial frequencies. But, it has to be taken into account that the amount of light power which is in the z-component decreases with the square of sinφ. Therefore, for sinφ=0.2 there is nearly no light power in the z-component. For increasing values sinφ, there is an increasing difference between the two curves of the MTF with spatial frequencies in x-and y-direction for the case of linear polarization. This is especially valid if the total electric energy density is taken, but also in a less pronounced form if only the y-component is taken.

For a final conclusion we have to distinguish between several cases:

4.2 Annular aperture with inner radius equal to 90% of the aperture radius

The MTF curves for the annular aperture in Fig. 2 show the remarkable effect that for all values of sinφ the curves of linear polarization with spatial frequencies in x-direction and using all electric field components (blue solid lines) nearly coincide with the curves of radial polarization using only the z-component (long dashed black lines). Only for sinφ=1.0 there is a small difference in both curves. An explanation for this similarity of the MTF curves is that the PSF along a central section in x-direction approaches the theoretical scalar PSF if the light is linearly polarized in y-direction and if an annular aperture is used. The same is valid for the axial component of the electric energy density in the case of radially polarized light and an annular aperture.

It can also be seen that for sinφ=0.2, which is for linear and circular polarization nearly equivalent to the scalar case, all curves with the exception of the curve for radial polarization using all electric field components (green short dashed line) coincide.

Similar to the case of a full circular aperture, there are for the annular aperture zeros of the MTF for radial polarization using all components of the electric field (green short dashed lines) up to values sinφ=0.7 and for linear polarization using all components of the electric field in the case of spatial frequencies in y-direction (blue dashed lines) if sinφ≥0.8.

If we have a detector which is only sensitive to the y-component of the electric field, the curves for linear polarization show for increasing values sinφ an also increasing difference between the contrast for structures with spatial frequencies in x- (red solid lines) and y-direction (red dashed lines). For sinφ=1.0 the contrast for spatial frequencies in x-direction can approach the very high value of about 0.175 for ν _x=1.88 NA/λ! On the other side, for spatial frequencies in y-direction with the same modulus, i.e. ν _y=1.88 NA/λ, the contrast is only 0.02, i.e. nearly ten times smaller as in the x-direction. If we consider for comparison the same curves for the full circular aperture at the same values of the spatial frequencies we see that there the contrast is only about 0.06 in x-direction and 0.01 in y-direction. So, the annular aperture is useful for the imaging of structures with spatial frequencies near the cut-off frequency, whereas it is not so useful for the imaging of small or medium spatial frequencies.

4.3 Apodization effects for radial polarization and sinφ=1.0

Finally, the apodization effects shall be discussed for the case of radial polarization and the limiting case sinφ=1.0 (Figs. 3–6).

For the annular apertures with different inner radii (case (i)) Fig. 3 shows that there is nearly no difference between the curve of the full aperture (r _in=0) and the curves with r _in/r _aperture<0.4, especially if all components of the electric field are detected. For increasing values r _in/r _aperture the contrast for small and medium spatial frequencies decreases whereas the contrast for very high spatial frequencies near the cut-off frequency increases by shaping a maximum of the MTF which approaches more and more the cut-off frequency. Of course, the existence of a maximum of the MTF for high ratios r _in/r _aperture at very high spatial frequencies means that there exists also a minimum of the contrast at medium spatial frequencies, whereas the MTF curve is strictly monotonic decreasing for r _in/r _aperture≤0.3.

Similar behaviors can be seen for the smoothly apodized pupils of the cases (iia) and (iib), but there the maxima for high spatial frequencies are less pronounced and the contrast is fairly high over a large range of spatial frequencies. Whereas Fig. 4, i.e. case (iia), shows some differences between the curves for very high spatial frequencies, there are nearly no differences for very high spatial frequencies in Fig. 5, i.e. case (iib). Totally, the curves in Fig. 5, i.e. if the maximum of the amplitude is at the rim of the aperture for all different amplitude functions, differ less than in Fig. 4, where a constant waist parameter w ₀=0.95 r _aperture is taken.

Finally, in Fig. 6 some curves of Figs. 3, 4, and 5 are combined in the same figure to show that nearly the same MTF curves can be obtained by using either a “hard mask” apodization via an annular aperture and homogeneous intensity or by using a smooth apodization via a certain smooth amplitude function. There, the amplitude functions |E̱| = r ³ exp(-r ² /w ² ₀) and w ₀=0.95 r _aperture or w ₀=0.82 r _aperture (i.e. maximum at the rim of the aperture) or |E̱| = r ⁴ exp(-r ² /w ² ₀) and w ₀=0.71 r _aperture (i.e. also maximum at the rim of the aperture) are compared to the annular aperture with r _in/r _aperture=0.6.

5. Application in lithography and microscopy

 

Fig. 7. Optical system for the imaging of an extended object which consists of small particles which emit like a dipole with its axis parallel to the optical axis.

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In this section we will discuss ideas on how to apply the results. In Ref. [4] it was proposed that particles oscillating like a dipole with its axis parallel to the optical axis will emit radially polarized light which can be captured by a high NA microscope objective. In the image plane of this microscope objective a magnified image will be formed and the PSFs of the single particles which are assumed to be incoherent to each other will show the typical form for radially polarized light. But, due to the magnification factor (β≪1, the numerical aperture NA_image will be demagnified by the same factor compared to the numerical aperture in the object space NA_Obj

provided the microscope objective fulfills the sine condition. In the image plane we find that different field components are distributed differently. On one hand there is a well focused but quite weak longitudinal component of the electric energy density. On the other hand we also find a strong and broad lateral component. The latter decreases the resolution if the detector is sensitive to the total electric energy density and does not distinguish between the lateral and longitudinal component (see for comparison the green dashed MTF curve in Fig. 1 for the small value sinφ=0.2). One solution would be to use a detector sensitive only to the longitudinal component, sacrificing most of the intensity. Another and better solution is using a polarization converting element introduced into the Fourier plane of the microscope objective, that converts radial polarization into linear polarization (see Fig. 7). This polarization converter may be combined with an apodizing element. This way a normal PSF of linearly polarized light will be formed in the image plane. As was shown in section 4 the MTF for linearly polarized light will have higher modulation than for radially polarized light if the numerical aperture is much smaller than 1 (see again Fig. 1 for the small value sinφ=0.2). Of course this application in microscopy only works if each particle in the object emits like a dipole with radial polarization. If the object emits linearly or circularly polarized light the polarization state should not be changed in microscopy.

On the other hand, the same optical system can be applied to optical lithography if it is used in the opposite direction (see Fig. 8). A mask which should be demagnified by a factor of normally β=0.2 or 0.25 is illuminated with linearly polarized light. Since the numerical aperture in the object space NA_Obj is in this case according to Eq. (7) smaller than or equal to β (assuming a non-immersion system with NA_image≤1), the plane wave behind the collimating lens will also be linearly polarized if polarization effects at the mask are neglected. Then, the polarization converting element forms a radially polarized mode which is focused by the high NA objective to a tight spot if the aperture angle sinφ is larger than 0.9. So, the modulation of the image will be increased for high spatial frequencies compared to using linearly polarized light which would have a zero of the MTF for grating-like structures with the grating lines perpendicular to the direction of polarization. Normally, polarization effects at the mask with periods down to p=1/(βν _cut)=2λ/NA (assuming β=0.25) cannot be neglected and locally elliptical polarized light will result behind the mask if the grating lines are oriented with an arbitrary angle relative to the direction of polarization of the incident light. But then, a polarizer in front of the polarization converting element can produce a well-defined linear polarization state without blocking too much light.

 

Fig. 8. Optical system for optical lithography at high aperture angles. The mask emits nearly linearly polarized light which is collimated by the first lens. The polarization converting element forms radially polarized light from the linearly polarized plane wave coming from one of the object points. The high NA objective behind the polarization converter focuses the light to a tight spot.

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Such a polarization converter was proposed in the context of a lithographic system in former patents [16, 17], but for quite a different purpose. In these patents the goal was to minimize the large angle Fresnel reflection losses at the photo resist and also in the lenses of the projection lens. Illumination with radial polarization leads to TM polarized light at all interfaces and for angles close to Brewster’s angle nearly all light is transmitted. Of course, this effect of radially polarized light is also an additional advantage in our proposal.

Unfortunately an efficient polarization converting element, which does not introduce any aberrations and also works for off-axis illumination does not exist. The reason that it has also to work for off-axis illumination is that off-axis object points cause tilted plane waves at the polarization converting element. But there are some demonstration elements based on an array of properly oriented half wave plates [17], liquid crystal devices [18] or zero order gratings [19-25]. A promising approach uses high frequency diffractive optical elements which act as an artificial birefringent medium [19, 20]. Such elements can be used as position dependent quarter or half wavelength plates [21–25]. One advantage such elements have over liquid crystal devices is that they work also with UV light, at least in principle. But the very small minimum feature sizes and quite large depths of the structures are not easy to fabricate.

6. Conclusion

We calculated the incoherent modulation transfer function for imaging with a diffraction-limited high numerical aperture optical system for different polarization states and different apodization at the pupil. We showed that the MTF can have zeros if all components of the electric energy density are detected, although in the scalar approximation there would be only zeros of the MTF for optical systems with aberrations.

We discussed the different MTF curves assuming that different types of detectors exist. These are either sensitive to the total electric energy density which is normally the case or are assumed to be sensitive to only one polarization component of the electric energy density. We studied the case of linear incident polarization in which only the component of the electric field parallel to the incident polarization is used. Grating-like structures with the grating lines parallel to the incident polarization are imaged with the best contrast for high spatial frequencies. But the contrast is much smaller for structures with grating lines perpendicular to the direction of polarization. On the other hand, radially polarized light yields a rotationally symmetric MTF. For high aperture angles, i.e. high sinφ with a value of nearly 1.0, the contrast is comparatively high. Thus, depending on the symmetry of the structures to be imaged different polarization states should be used: linear polarization for grating-like structures with only one orientation of the grating lines and radial polarization for structures with arbitrary orientations. At this point it may be interesting to note that at high spatial frequencies the MTF calculated in the vectorial theory (s. Figs. 1 and 2) can be higher than the limiting MTF obtained for the unrealistic assumption of scalar fields (for comparison see the curves in Figs. 1 and 2 for linear and circular polarization and sinφ=0.2 which correspond very well to the scalar MTF curve). A detector which is only sensitive to a certain electric field component is required for exploiting this advantage.

Finally we proposed the principle design of a system which can apply these results to lithography by converting the polarization state between the object and the image space. By specifically controlling the polarization state an increase of the resolution should be achievable, even in the case of the incoherent imaging of a whole object field.

7. Appendix: Calculation of the PSF of an aplanatic fast lens

The PSF is calculated using a numerical implementation of the Debye integral similar to Ref. [1] or Refs. [2, 3]. The principal idea is that the electric field in the vicinity of the focus can be calculated by superimposing plane wave components which propagate along the direction of the geometrical rays to the focus. Of course, by doing this the local polarization, local phase and local amplitude of the plane wave components have to be taken into account correctly. In the entrance pupil of the aplanatic lens the electric field E is numerically sampled by taking an array of equidistant rays which represent local plane wave components. Each ray number j is associated with a polarization vector P _j which is a complex valued vector perpendicular to the direction of propagation e _j of the ray. Its modulus |P _j| is proportional to the local electric field E if the rays are equidistant and its direction represents the direction of polarization for locally linearly polarized light. OPD_j is the optical path length of the ray in the entrance pupil. It is zero for a real plane wave without aberrations.

For an infinite distant object point (i.e. incident plane wave) the aplanatic lens now deflects each ray with the height$h=\sqrt{x^2+y^2}$ in the entrance pupil in such a way that it seems to come from a sphere (exit pupil) with radius f around the focus fulfilling the sine condition (see Fig. 9):

 

Fig. 9. Scheme of a lens fulfilling the sine condition.

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Here, ϑ is the angle of the deflected ray with the optical axis (z-axis). In the focus all these rays will have the same optical path length as in the entrance pupil (i.e. zero for an incident plane wave without aberrations). The deflected rays now propagate along new direction vectors e’_j pointing from the exit pupil (sphere) to the focus and the new polarization vectors P’_j are calculated in such a way that the component perpendicular to the plane of deflection remains the same and the component in the plane of deflection is rotated so that it is perpendicular to the new direction vector e’_j. The equations which are simple vector calculus will not be shown here.

The only thing which has to be considered carefully is that the modulus |P’_j| of the polarization vector of the deflected rays is different from the modulus |P _j| of the incident rays so that there is a scaling g(ϑ) for each polarization vector:

This is a consequence of energy conservation. To calculate the factor g it has to be taken into account that the rays in the exit pupil are no longer equidistant and therefore the amplitude A’ (which is proportional to the modulus of the electric field) associated with each plane wave component along a ray is given by A′= |P′|/dF′ . Here, dF’ is the surface area element in the exit pupil which each ray covers in the numerical sampling. In the plane entrance pupil the surface area elements dF are equal for all rays since the rays were sampled equidistant. However, in the exit pupil, which is for an aplanatic lens a sphere around the focus, the surface area elements change geometrically by dF′=dF/cosϑ (see Fig. 9). The amplitude A’ of the plane wave component in the exit pupil is connected to the amplitude A of the plane wave component in the entrance pupil by the well-known factor [1] A′= A√cosϑ. Therefore, the factor g is in total:

Finally, the electric field in the focus is calculated by:

The parameter α is a constant of proportionality which can be set to 1 if we are only interested in relative units of E. Here, the summation is done over all rays/plane wave components in the exit pupil and as mentioned before for a plane wave without aberrations the optical path lengths OPD_j are zero. However, also aberrations of the incident wave can simply be taken into account with our method. Another advantage is that the aperture shape can be arbitrary (circular, annular, rectangular, etc.) by just changing the rays which are summed up. If other focusing elements are taken just the parameter g has to be changed and the new direction vectors e’ and polarization vectors P’ have to be calculated accordingly. For reflection at a parabolic mirror the factor g is for example just constant and equal to 1 because by reflection the cross-section and amplitude of a plane wave component is not changed.

We checked of course our numerical method by comparing our results of the PSF with those of authors [4–8, 26] taking directly the Debye integral of Ref. [1]. The methods give the same results if a sufficiently high number of plane wave components (rays) is taken like for example 100×100 plane wave components for sampling the entrance pupil.